"Lack of [something]" is specific enough to use some kind of article, but we often leave it off for style. All of these are fine:
The event was canceled due to lack of interest.
The event was canceled due to a lack of interest.
The event was canceled due to the lack of interest.
The event was canceled due to their lack of interest.
In this way "lack" is similar to many other English nouns that relate to abstract concepts. Examples:
Strength of purpose often distinguishes the successful from the unsuccessful.
Compassion for those in need characterized her personality.
Each of these could take an article, but leaving it off sometimes sounds better, as it implies these are universal concepts.
In the same way "lack of money" is a universal problem and does not need an article. Still, with due respect to Cambridge, I would have written your first sentence as:
... because of a lack of money.
The money to pay hospital departments comes from a specific source, and so it's a little misleading to suggest it is an abstract need. There's nothing really wrong with the Cambridge version, but I disagree with the nuance.